I intend to give an introduction to the work on boundary values, done over the last few years in collaboration with E.L. Stout.

Recall a classical result. If $u$ is a harmonic function defined on the unit ball in ${\bf R}^n$, then it is possible to define a boundary value, along the unit sphere. In case $u$ is bounded one can naively use radial limits. But, in fact, much easier is the theory of functions $u$ with moderate growth (so-called polynomial growth), or unrestricted growth. One then gets boundary values that are distributions, or respectively analytic functionals (using then only real analytic test functions). The study of similar problems along non compact boundary immediately leads to the notion of hyperfunction (First exercise: if $u$ is a holomorphic function in the upper half plane, what is its boundary value along the interval $[-1,+1]$?).

Our work has been motivated by a fundamental question: when is it legitimate to speak about the boundary value of a function? We have developed a general theory in the real analytic setting. This theory includes the theory of boundary values for solutions of P.D.E along non-characteristic boundaries, and it clarifies it.

I hope to explain the main points of our work, and also to point out some unsolved questions (including the mysterious question of a theory in the smooth setting).

Having to deal with limits (boundary values), and with the question of carriers (unavoidable, when using analytic functionals for defining hyperfunctions), we had to face difficulties that are already illustrated by the following simple example. For each $n$, the distribution, on ${\bf R}$, $\varphi \mapsto \sum_{k=0}^n{\varphi^{(k)}(0) \over k!}$, is carried (supported) by $\{0\}$. But, these distributions converge, in the appropriate sense, to the functional $\varphi \mapsto \varphi (1)$, certainly not carried by $\{0\}$. The only proof that I shall try to sketch, will be a proof of the classical theorem of Paley-Wiener, following a new approach. It leads to a non linear Paley-Wiener Theory, which is crucial to handle the delicate but absolutely crucial question of carriers of limits.